lecture 10 the poisson distribution math 1107 introduction to statistics

Lecture 10The Poisson Distribution

MATH 1107Introduction to Statistics

Math 1107 Poisson Distribution

What if we are interested in obtaining the probability of a “success” when the number of “failures” is potentially infinite? Such as the probability of a web site hit? Or the probability of being in a car accident?

A poisson probability distribution results from a procedure that meets all the following requirements:

Math 1107 Poisson Distribution

The random variable x is the number of occurrences of an event over some interval.

The occurrences must be random.

The occurrences must be independent of each other.

The occurrences must be uniformly distributed over the interval being used.

Math 1107 Poisson Distribution

P(x) = µ x • e -µ

x!

where e 2.71828µ = average number of occurrencesX is the occurrence of interest

The Poisson distribution differs from the binomial distribution in these fundamental ways:

• The binomial distribution is affected by the sample size n and the probability p, whereas the Poisson distribution is affected only by the mean μ.

• In a binomial distribution the possible values of the random variable are x are 0, 1, . . . n, but a Poisson distribution has possible x values of 0, 1, . . . , with no upper limit.

Math 1107 Poisson Distribution

Math 1107 Poisson Distribution

World War II Bombs In analyzing hits by V-1 buzz bombs in World War II, South London was subdivided into 576 regions, each with an area of 0.25 km2. A total of 535 bombs hit the combined area of 576 regions

If a region is randomly selected, find the probability that it was hit exactly twice.

The Poisson distribution applies because we are dealing with occurrences of an event (bomb hits) over some interval (a region with area of 0.25 km2).

The mean number of hits per region is

μ number of hits

number of regions

5350.929

576

The probability of a particular region being hit exactly twice is P(2) = 0.170.

Math 1107 Poisson Distribution

P(x) = .9292 • e -929

2!

Fun EXCEL Exercise

Example 2 –

For a period of 100 years, there were 93 major earthquakes in the world. What is the probability that the number of earthquakes in a randomly selected year is 5?

Math 1107 Poisson Distribution

P(x) = .93 5 • e -.93

5! .00229=

Example 3 –

A certain machine process generates 1 defect for every 200 units produced per day. What is the probability of generating exactly 3 defects in a single day?

Math 1107 Poisson Distribution

P(x) = .0023 • e -.002

3! .00000000207=

Example 4 –

You work for a large property insurance company in Florida. You need to determine the needed cash reserves for the upcoming hurricane season. You know that in the last 52 years, Florida has been hit with 72 hurricanes. Each hurricane generates approximately $10M in claims. What is the probability that this year, Florida will experience 3 hurricanes?

Math 1107 Poisson Distribution

P(x) = 1.3853 • e –1.385

3! .11084=